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For a commutative ring $R$, I can consider $\operatorname{GL}_n(R)$ as a group scheme over $\operatorname{Spec} R$. Are there analogs of this notion when $R$ is non-commutative, say $R = \operatorname{End}_k V$ (for $V$ a finite-dimensional $k$-vector space), which retain any useful part of the theory of group schemes?

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Well, it's easy to define a group-valued functor on the category of not-necessarily-commutative rings. But group schemes in the usual sense are very special group-valued functors, and it is not so easy to generalise that part of the definition to the non-commutative setting. – Zhen Lin Nov 3 '12 at 1:16
I know it's tough; that's why I asked. I was thinking in terms of modern notions of spec of a noncommutative ring (which I don't really understand). – Daniel McLaury Nov 3 '12 at 4:16

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